For every positive integer N and α∈[0,1), let B(N,α) denote the probabilistic model in which a random set A⊂{1,…,N} is constructed by choosing independently element of {1,…,N} with probability α. We prove that, as N⟶+∞, for A we have |AA|∼|A|2/2 1−o(1), if only iflog⁡(α2(log⁡N)log⁡4−1)log⁡log⁡N⟶−∞. This improves on theorem Cilleruelo, Ramana Ramaré, who proved above asymptotic between |AA| |A|2...

In this work, we shall present some novel process to measure the similarity between picture fuzzy sets. Firstly, we adopt the concept of intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets and picture fuzzy sets. Secondly, we develop some similarity measures between picture fuzzy sets, such as, cosine similarity measure, weighted cosine similarity measure, set-theoretic similar...

In this paper, we study both concepts of geodetic dominatingand edge geodetic dominating sets and derive some tight upper bounds onthe edge geodetic and the edge geodetic domination numbers. We also obtainattainable upper bounds on the maximum number of elements in a partitionof a vertex set of a connected graph into geodetic sets, edge geodetic sets,geodetic domin...